Is gradient descent scale invariant or not?
What are you hoping to get out of the answer for this question?
Feature scaling is a method you CAN(but don't have to) use, so that your algorithm performs faster and reaches better general accuracy.
I would say that on a simple regression task, where the feature value ranges do not vary a lot, the output would probably be almost the same, but as soon as you introduce data that has one feature of range 1-10, and other range of 10000-100000, that is where you would notice that you NEED to standardize/normalize your features in order to reach optimal results.
That's why it's almost a general rule to just scale your data, so your algorithm can generalize better, and you don't have to worry about your algorithm giving higher importance to a feature with higher values, instead of the one with lower values(just an example).